The first-column inverse-matrix partition-function conjecture

Let $p(n)$ denote the partition function, let $\mu(n)$ denote the Möbius function, and let $s_{n,1}^{(-1)}$ denote the entries in the first column of the inverse matrix under consideration. First-column inverse-matrix conjecture. The first column is given by the convolution

$$s_{n,1}^{(-1)}=\sum_{d\mid n}p(d-1)\mu(n/d).$$

Equivalently, Möbius inversion gives

$$p(n-1)=\sum_{d\mid n}s_{d,1}^{(-1)}.$$

The source presents this as an experimental conjecture based on the observed initial values and an OEIS match; no proof or resolution is supplied.